A Power-Law Upper Bound on the Correlations in the 2D Random Field Ising Model

Abstract

As first asserted by Y. Imry and S-K Ma, the famed discontinuity of the magnetization as function of the magnetic field in the two dimensional Ising model is eliminated, for all temperatures, through the addition of quenched random magnetic field of uniform variance, even if that is small. This statement is quantified here by a power-law upper bound on the decay rate of the effect of boundary conditions on the magnetization in finite systems, as function of the distance to the boundary. Unlike exponential decay which is only proven for strong disorder or high temperature, the power-law upper bound is established here for all field strengths and at all temperatures, including zero, for the case of independent Gaussian random field. Our analysis proceeds through a streamlined and quantified version of the Aizenman–Wehr proof of the Imry–Ma rounding effect.

Appendices

Appendix A. Exponential Decay at High Disorder

As a rule of thumb it is generally expected that at high enough disorder, be it thermal or due to noisy environment, correlations decay exponentially fast. Results in this vein for systems related to the RFIM can be found in the works of Berretti [6], Imbrie and Fröhlich [16], and Camia et al. [9].

Let us present here an especially simple proof of such behavior for the $T=0$ case, i.e. exponential decay of the correlations of the RFIM’s ground state, and also of the principle that fast enough power-law decay implies exponential decay.

Theorem A.1

For the RFIM on $\mathbb {Z}^d$ with the nearest-neighbor interaction (1.5) and random field given by IID random variables $(\eta _u)$, if

$$\begin{aligned} \mathbb {P}\big (|h+\varepsilon \eta _\mathbf 0 | \le 2d J \big ) \ < \ p_c(d) \, \end{aligned}$$

(A.1)

with $p_c(d)$ the critical density for site percolation on $\mathbb {Z}^d$, then $m(L;0,\mathcal {J}, h,\varepsilon )$ decays exponentially fast in L.

Proof

At sites where $|h+\varepsilon \eta _v| > 2d J$ the ground-state configuration is dictated by the sign of the local field. Hence disagreement percolation can propagate only along the sites with $|h+\varepsilon \eta _v| \le 2d J$. In the regime described by (A.1) the exceptional sites form a sub-percolating point process, for which the connectivity probability is known to decay exponentially in the distance [2, 19]. $\quad \square$

A boosted version of the above simple argument allows to conclude that if on some scale $\ell$ the probability of influence propagation is small enough ($1/\ell ^{d-1}$ power law with a small prefactor) then on larger scales the influence decays exponentially fast. An analogous statement holds also for $T>0$, but for simplicity of presentation we present the proof for $T=0$.

Theorem A.2

For the RFIM on $\mathbb {Z}^d$ with the nearest-neighbor interaction (1.5), there is a finite constant $c_0$ (depending only on d) with which: if for some $\ell < \infty$

$$\begin{aligned} m(\ell ;0,\mathcal {J}, h,\varepsilon ) \, \le \, c_0 / \ell ^{d-1} \end{aligned}$$

(A.2)

then for all $L <\infty$

$$\begin{aligned} m(L;0,\mathcal {J}, h,\varepsilon ) \, \le \, C_1 \, e^{- b L/ \ell } \end{aligned}$$

(A.3)

with $C_1, b \in (0,\infty )$ which do not depend on J, h, $\epsilon$ and $\ell$.

In particular, we learn that $m(L;0,\mathcal {J}, h,\varepsilon )$ cannot decay by a power law faster than 1 / L without decaying exponentially.

Proof

In the following we say that a site $v\in \mathbb {Z}^2$ is sensitive to boundary conditions at distance $\ell$ if

$$\begin{aligned} \sigma ^{\Lambda _v(\ell ),+}_v \ne \sigma ^{\Lambda _v(\ell ),-}_v. \end{aligned}$$

(A.4)

For each $L> \ell$ the event whose probability defines m(L) ,

$$\begin{aligned} \langle \sigma _\mathbf 0 \rangle ^{\Lambda (L), +} \ \ne \ \langle \sigma _\mathbf 0 \rangle ^{\Lambda (L), -}, \end{aligned}$$

(A.5)

requires the existence of a path from $\mathbf 0$ to the set $\partial _{{\mathrm{v}}}\Lambda (L-\ell )$ along sites $v\in \mathbb {Z}^2$ at which the condition (A.4) holds.

Let now $\mathcal {P}_\ell$ be a partition of the vertex set of $\mathbb {Z}^d$ into a $\mathbb {Z}^d$-like array of disjoint cubic blocks of side length $2\ell$, and consider the random set of blocks in this partition which contain at least one site for which (A.4) holds. These block events are 1-step independent, in the sense that they are jointly independent for any collection of blocks of which no two are touching.

The probability that an individual block contains a site at which the condition (A.4) holds is trivially dominated by $|\partial _{{\mathrm{v}}}\Lambda (\ell )| \times m(\ell )$. Adjusting the constant $c_0$ in assumption (A.2) the above probability can be made as small as convenient. The claim then follows through a standard exponentially-decaying bound on the connectivity probability in 1-step independent percolation of small enough density. $\quad \square$

Appendix B. The Mandelbrot Percolation Analogy

The results presented above do not answer the question whether in two dimensions the exponential decay of correlations persists into arbitrarily small values of the disorder parameter, or whether the exponential decay turns into a power-law decay at low enough (but still non zero) $\varepsilon$. Related to this is the question of what would be a sensible algorithm for the computation of the ground state $\widehat{\sigma }$ for a given random field, and how would it perform at very low disorder.

An intriguing perspective is provided by the following hierarchal algorithm. It has the virtue of simplicity but also the drawback of being potentially misleading through over simplification. It is formulated for the specific case $h=0$ and nearest-neighbor interaction.

Let $(\mathcal {P}_n)$, $n\ge 0$, be a sequence of nested partitions of $\mathbb {Z}^2$ into square blocks, with the blocks in $\mathcal {P}_n$ having side-length $3^n$ and the square containing x denoted by $D_{n,x}$. For each n, x we define the following as a large-field event in $D_{n,x}$:

$$\begin{aligned} \mathcal {F}_{n,x}\,{:}{=}\, \{ \eta \, : \ \varepsilon \, \big |\eta (D_{n,x}) \big | \ > \ J \, |\partial _{\mathrm{e}} D_{n,x}| \}. \end{aligned}$$

(B.1)

where $\eta (D) \,{:}{=}\, \sum _{x\in D} \eta _x$ is the total block field.

A relevant feature of two dimensions is that the probabilities of the large-field events are scale invariant:

$$\begin{aligned} \mathbb {P}(\mathcal {F}_{n,x}) \ = \ \chi (4J/\varepsilon )\,{:}{=}\,p \ \approx \exp [-8J^2/\varepsilon ^2]. \end{aligned}$$

(B.2)

For a given $x\in \mathbb {Z}^2$ the events $\mathcal {F}_{n,x}$ are not strictly independent, however the sequence (in n) of the corresponding indicator functions is easily seen to be asymptotic, in probability, to a stationary and mixing sequence of random variables.

Let $n(x; \eta )$ be the first non-negative integer for which large field is exhibited in $D_{n,x}$. Due to the above properties of the events $\mathcal {F}_{n,x}$ for any $J,\varepsilon >0$ almost surely $n(x; \eta ) < \infty$ for all x.

Under $\mathcal {F}_{0,x}$, i.e. in case the large-field event occurs at x already on the smallest scale, the value of the ground-state configuration at x is predictably given by ${\mathrm{sign}} (\eta _x)$, i.e. the sign of the field. In case the $\eta _x$ is itself not large enough to meet this criterion, but the site is separated from the boundary of a set $\Lambda$ by a loop of sites for which the large-field events occur at scale $n=0$, one may still conclude that the finite-volume ground state at x does not depend on the boundary spin configuration $\sigma _{\partial _{{\mathrm{v}}}\Lambda }$.

Scaling up these observations, though along the way departing from rigor, we arrive at the following somewhat over-simplified algorithm for the assignment of a spin configuration $\tau (\eta )$ which may mimic the infinite-volume ground state $\widehat{\sigma }(\eta )$.

For each $x\in \mathbb {Z}^2$ let $k(x; \eta )$ be defined as the smallest $0\le k< n(x; \eta )$ for which x is separated from infinity by a loop of sites with $n(x; \eta ) \le k$, if such a k exists, and otherwise set $k(x; \eta )=n(x; \eta )$.

In the first case, i.e. $n(x; \eta ) = k(x;\eta )$, we let $\tau (\eta )_x = {\mathrm{sign} } (\eta (D(x))$. If $k(x;\eta ) < n(x; \eta )$, the value of $\tau (\eta )_x$ is determined by minimizing the RFIM energy over the interior of the corresponding x-encapsulating loop, with the previously constructed values serving as boundary conditions for $\tau (\eta )_x$.

For the finite-volume version of the construction, in $\Lambda \subset \mathbb {Z}^2$, the above construction is modified by limiting the considerations of large-field events to cubes contained in $\Lambda$. In the last step, unless $\tau ^{\Lambda ,\pm }(\eta )_x$ is defined already through such events, its calculation will incorporate the boundary conditions imposed at $\partial _{{\mathrm{v}}}\Lambda$.

Under the above algorithm the influence of the boundary conditions on $\tau (\eta )_x$ percolates over sites for which the events $\mathcal {F}_{n,x}$ did not yet occur. For an idea on the probability that the influence percolates deep inside $\Lambda$ one may take the further approximation in which the correlations between the indicator functions of nested events $\mathcal {F}_{n,x}$ are ignored.

Under the latter approximation, the collection of sites not covered by any of the large-field events, has the distribution of the random fractal set discussed in Mandelbrot’s “canonical curdling” model [18]. In particular, the influence-percolation process coincides with the Mandelbrot-percolation process at density p given by (B.2).

Curiously, as was proven by Chayes–Chayes–Durrett  [11], the Mandelbrot-percolation process does undergo a phase transition. Its manifestation in the lattice version of the model is that the connectivity function decays exponentially fast for p large enough, but at p small the decay changes to a power law. (The model is most appealing in its continuum, or “ultraviolet”, limit while our discussion is focused on its infinite-volume, or “infrared”, limit. However in the analysis there is a simple relation between the two).

It should however be noted that for the finite-volume version of the construction, the existence of a path connecting x to $\partial _{{\mathrm{v}}}\Lambda$ in the complement of the set of sites covered by large-field events is only a necessary condition for the dependence of $\tau ^{\Lambda ,\pm }(\eta )_x$ on the boundary conditions. As its value is determined through the energy minimization conditioned on both the ± boundary conditions and the randomly determined values along the large-field sets, the ± boundary conditions may lose their effect on $\tau (\eta )_x$ even before the geometric disconnection of x from $\partial _{{\mathrm{v}}}\Lambda$. Thus the Mandelbrot-percolation’s phase transition does not preclude exponential decay of the $\tau$-analog of our finite-volume order parameter at all $p>0$.